We just bought a house in September and this week we have had chilly temps. Well turned on the heat and Furnace pilot light went out, we re lit it and after the heat went off the flame did also. Next day wake up and we have no hot water. The pilot light on the water heater went out called a plumber he came out and said nothing was wrong with the heater and re lit it within a couple of hours pilot light was out. We have placed a service call to a HVAC tech and are waiting to see what he will find. Could this be a gas line issue.HELP.

Yes could be a gas line issue.
Could be a combustion air problem.
Could be a chimney down draft problem.
Could be spider web/dirt in the pilot oriface problem.

Could be the drip leg is full. They are the little capped pipe down to nowhere. Shut the gas off, remove it, and empty any water out. Replace and try again.

The pilot light on the water heater will not stay lit, any solution

We left a bad thermocouple out above. Plenty of good ones get replaced because they are cheap and fairly easy to replace. If that doesn't do the job, then we look elsewhere. The other thing is to make sure the thermocouple is in the pilot light flame.